CAME lesson 01 - Roofs

Learning Objective:

• we are learning to use algebra to solve problems

Investing:

• The whole point of the lesson is to show you why algebra can be useful to describe things

Preparing:

• Here is some triangular dotty paper:

• Note that the lines of dots run across the paper, not up and down.
• When you print the paper below onto A4, neighbouring dots should be exactly 1cm apart.
• Download-able paper:
  
  

  triangular spotty paper landscape.docx
  

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  triangular spotty paper landscape.pdf
  

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Modeling:

• Here is an example of a roof:
  
• You could call it an isosceles trapezium if you want to be precise, but 'roof' will do for now.
• You can draw more shapes on this Word document using the Review>Inking tools if you like:
  
  
  

  roofs examples.docx
  

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• Here is an examples of a not-roof:
  
• A shape fails to be a roof if the lengths don't make a single closed isosceles trapezium.
• Watch out for times when there is an overlap or the shape isn't closed.
• Notice how the numbers tell you about the shape: each describes the length of the four sides starting from the bottom left and going clockwise.
• All our shapes will have four numbers and the directions of our lines will stay the same.

Discovering:

• Let's play with some roofs and not-roofs.
• You should draw yours on dotty paper, but we can check our drawings with this Desmos widget:

• Draw a shape using the lengths 1, 2, 1, 3. Is it a roof or a not-roof?
• Draw a shape using the lengths 2, 2, 2, 4. How about this one? is it a roof or a not-roof?
• Now try 2, 1, 2, 2. Roof or not?
• 3, 1, 3, 4?
• 2, 3, 1, 4?

• Draw any new roof you like. Write down its number code.
• Draw your own not-roof. Write down its code.

• Design another roof. This time write down the code before you draw it. How did you know it would work?
• Design another not-roof. Again write down the code before you draw. What about the code told you it would fail?
• Test your ideas by drawing another example of a roof.
• Test your ideas by drawing another example of a not-roof.

• Write down (in words) a rule you have noticed about the numbers.
• Write down another rule.

Discussing:

• Discuss your rules with a learning partner.
  • maybe discuss them as a class?
    
    
• Using your rules, what would 17, 13, 17, 30 be?
  • would it be a roof or not?
  • how can you tell?
  • tell your learning partner; convince them you're right.
  • explain how you know.
    
    
• What would 14, 11, 15, 26 be?
  • why?
    
    
• How about 19, 10, 19, 26?
  • how could you be sure without drawing it?
    
    
• Let's try your rules out on my Desmos widget:

• Notice that you can move the sliders to change the values for the lengths.
• Notice I've given the sliders names: a, b, c and d.
• How can you re-frame your rules using my letters?
  • can you tell me a rule about the first and third length in only three symbols?
  • how can you write your rules about the first few lengths and the last length in as short a way as possible?
  • does anyone in the class have a different rule?
  • can we re-write these rules in a different way?

Explaining:

• One way to do this is...
• Another approach might be...
• A useful shortcut is to...
• This works because...
• It doesn't work when...
• An exception is...
• Watch out for...
• A common mistake is...
• You can check your result by...
• We can prove this works by...

Practicing:

• Some straightforward examples.
• Some harder examples.
• Some mixed examples.
• Some non-examples to spot and some mixed questions with redundant, insufficient or contradictory data.
• You can demonstrate fluency by at least...

Sharing:

• A web page or wiki I have created to explain this can be found at...
• A presentation I have created and rehearsed looks like...
• A poster I have drawn or model I have made can be found...

Assessing:

• Check you've mastered this skill by...
• Show you understand by explaining...
• Prove you're an expert in... by...

Developing:

• Next we could learn...
• This leads to...
• Now try...